Next, we show that for any given \(\epsilon _2>0\) , \(H(X^{n}|W_1,Y_2^{n},U^{n})\le n\epsilon _2\)  for large enough \(n\) . To calculate \(H(X^{n}|W_1,Y_2^{n},U^{n})\) ,
consider the following hypothetical scenario. Fix \(W_1 = w_1\) , and assume that
the transmitter sends a codeword \(x^{n}\bigl (w_1,j_1,u^{n}(w_2,j_2)\bigr )\) ,
\(j_1\in \lbrace 1,\dots ,J_1\rbrace \) . Assume that receiver 2 knows the sequence \(U^{n} =u^{n}(w_2,j_2)\) . Given index \(W_1 = w_1\) , receiver 2 decodes the codeword
\(x^{n}(w_1,j_1,u^{n})\)  (i.e., looks for the index \(j_1\) ) based on the received
sequence \(y_2\) . Let \(\lambda (w_1)\)  denote the average probability of error of
decoding the index \(j_1\)  at receiver 2. By the AEP [1]},
we have \(\lambda (w_1)\le \epsilon \)  for sufficiently large \(n\) . By Fano's
inequality [1]},
\(\frac{1}{n} H(X^{n}|W_1=w_1,Y_2^{n},U^{n})&\le \frac{1}{n}+\lambda (w_1)\frac{\log _2 J_1}{n}\\&\le \frac{1}{n}+\epsilon R^{\prime }_1\\&:= \epsilon _2.\) 
